Question: Suppose $f$ is a linear function. If $f(x+3) + f(4x+5) = 10x$, then $f(x)=?$
My attempts: Okay, this is what I have so far... $$f(x) + f(3) + 5(4x) + f(5)$$
$$f(5x) = 10x - f(8)$$
$$f(5x) = 2(5x) - f(8)$$
How do I simplify further to $f(x)$?
On
First I want to note that JimmyK4542 gave a really good constructive answer, and mine is not nearly as constructive.
$x=0\Rightarrow f(3)+f(5)=0$, but as $f$ is linear, $f(4)=0$, therefore $f(x)=a(x-4)$
For the next part, keep in mind that for any linear function $f$, $f(x)=ax+b$ we get that for any $c$, $f(x+c)-f(x)=ac$
Denote $y_0=x_0+3,y_1=4x_0+5$, then $f(y_0+1)+f(y_1+4)=10(x_0+1)=f(y_0)+f(y_1)+10$, therefore $a+4a=10\Rightarrow a=2$
For conclusion, $f(x)=2(x-4)=2x-8$
Lets check it: $f(x+3)+f(4x+5)=2x+6-8+8x+10-8=10x$ as required.
Try letting $f(x) = ax+b$ for some constants $a,b$. Then $f(x+3) = a(x+3)+b$ and $f(4x+5) = a(4x+5)+b$. Now, just solve for $a,b$.